We give an improved lower bound for the average of the Erd\H{ o}s-Hooley function $\Delta(n)$, namely $\sum_{ n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{ 1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of $\gg x \log\log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of $x (\log\log x)^{ 11/4}$ of the second and third authors.
Categories: math.NT